Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to Levy Characterization of Brownian motion. Thanks
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Certainly; if $M_n$ is a discrete-time $L^2$ martingale, then its quadratic variation $\langle M \rangle_n$ is the unique predictable increasing process such that $\langle M \rangle_0 = 0$ and $M_n^2 - \langle M \rangle_n$ is a martingale. The existence and uniqueness follow from Doob's decomposition, the discrete-time (and much simpler) precursor to the Doob-Meyer decomposition.
Simple random walk is not uniquely characterized by its quadratic variation; indeed, if $X_i$ are iid with any distribution having mean 0 and variance 1, then $M_n = X_1 + \dots + X_n$ is a martingale with quadratic variation $\langle M \rangle_n = n$.
answered Oct 19, 2013 at 3:08
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$\begingroup$ Has the convergence of the quadratic variation of discrete processes to the quadratic variation of continuous processes been studied, with some well known results? Discrete martingales can converge in distribution to their continuous analogues, I wonder if some of the properties "converge" too, i.e. the quadratic variation. $\endgroup$ Commented Mar 1 at 17:13
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1$\begingroup$ @JanStuller: Well, there's the Burkholder-Davis-Gundy inequality which implies that a sequence of (local) martingales converges uniformly in $L^p$ iff the quadratic variations converge in $L^{p/2}$. Not quite what you asked, but sort of in the same direction. I don't know off the top of my head about other results. $\endgroup$ Commented Mar 1 at 17:22
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There is also the quadratic variation, defined as the sum of the square of jumps (as in continuous time, you have, for discontinuous martingales the covariation process and the predictable covariation process)
